3.72 \(\int \frac{(a+b x)^4}{c+d x^3} \, dx\)

Optimal. Leaf size=282 \[ \frac{2 a^2 b^2 \log \left (c+d x^3\right )}{d}-\frac{\left (b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )-\sqrt [3]{d} \left (4 a b^3 c-a^4 d\right )\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{5/3}}+\frac{\left (b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )-\sqrt [3]{d} \left (4 a b^3 c-a^4 d\right )\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{5/3}}+\frac{\left (a^4 \left (-d^{4/3}\right )-4 a^3 b \sqrt [3]{c} d+4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{5/3}}+\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d} \]

[Out]

(4*a*b^3*x)/d + (b^4*x^2)/(2*d) + ((b^4*c^(4/3) + 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^
(1/3)*d - a^4*d^(4/3))*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[
3]*c^(2/3)*d^(5/3)) + ((b*c^(1/3)*(b^3*c - 4*a^3*d) - d^(1/3)*(4*a*b^3*c - a^4*d
))*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(5/3)) - ((b*c^(1/3)*(b^3*c - 4*a^3*d)
- d^(1/3)*(4*a*b^3*c - a^4*d))*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/
(6*c^(2/3)*d^(5/3)) + (2*a^2*b^2*Log[c + d*x^3])/d

_______________________________________________________________________________________

Rubi [A]  time = 0.796534, antiderivative size = 280, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{2 a^2 b^2 \log \left (c+d x^3\right )}{d}+\frac{\left (a^4 (-d)-\frac{b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )}{\sqrt [3]{d}}+4 a b^3 c\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}+\frac{\left (b \sqrt [3]{c} \left (b^3 c-4 a^3 d\right )-\sqrt [3]{d} \left (4 a b^3 c-a^4 d\right )\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{5/3}}+\frac{\left (a^4 \left (-d^{4/3}\right )-4 a^3 b \sqrt [3]{c} d+4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{5/3}}+\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4/(c + d*x^3),x]

[Out]

(4*a*b^3*x)/d + (b^4*x^2)/(2*d) + ((b^4*c^(4/3) + 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^
(1/3)*d - a^4*d^(4/3))*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[
3]*c^(2/3)*d^(5/3)) + ((b*c^(1/3)*(b^3*c - 4*a^3*d) - d^(1/3)*(4*a*b^3*c - a^4*d
))*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(5/3)) + ((4*a*b^3*c - a^4*d - (b*c^(1
/3)*(b^3*c - 4*a^3*d))/d^(1/3))*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/
(6*c^(2/3)*d^(4/3)) + (2*a^2*b^2*Log[c + d*x^3])/d

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a^{2} b^{2} \log{\left (c + d x^{3} \right )}}{d} + \frac{4 a b^{3} x}{d} + \frac{b^{4} \int x\, dx}{d} - \frac{\sqrt{3} \left (a \sqrt [3]{d} \left (a^{3} d - 4 b^{3} c\right ) + b \sqrt [3]{c} \left (4 a^{3} d - b^{3} c\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} d^{\frac{5}{3}}} + \frac{\left (a^{4} d^{\frac{4}{3}} - 4 a^{3} b \sqrt [3]{c} d - 4 a b^{3} c \sqrt [3]{d} + b^{4} c^{\frac{4}{3}}\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} d^{\frac{5}{3}}} - \frac{\left (a^{4} d^{\frac{4}{3}} - 4 a^{3} b \sqrt [3]{c} d - 4 a b^{3} c \sqrt [3]{d} + b^{4} c^{\frac{4}{3}}\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} d^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4/(d*x**3+c),x)

[Out]

2*a**2*b**2*log(c + d*x**3)/d + 4*a*b**3*x/d + b**4*Integral(x, x)/d - sqrt(3)*(
a*d**(1/3)*(a**3*d - 4*b**3*c) + b*c**(1/3)*(4*a**3*d - b**3*c))*atan(sqrt(3)*(c
**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(2/3)*d**(5/3)) + (a**4*d**(4/3) - 4
*a**3*b*c**(1/3)*d - 4*a*b**3*c*d**(1/3) + b**4*c**(4/3))*log(c**(1/3) + d**(1/3
)*x)/(3*c**(2/3)*d**(5/3)) - (a**4*d**(4/3) - 4*a**3*b*c**(1/3)*d - 4*a*b**3*c*d
**(1/3) + b**4*c**(4/3))*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*
c**(2/3)*d**(5/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.49217, size = 277, normalized size = 0.98 \[ \frac{12 a^2 b^2 d^{2/3} \log \left (c+d x^3\right )-\frac{\left (a^4 d^{4/3}-4 a^3 b \sqrt [3]{c} d-4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{2/3}}+\frac{2 \left (a^4 d^{4/3}-4 a^3 b \sqrt [3]{c} d-4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{2/3}}+\frac{2 \sqrt{3} \left (a^4 \left (-d^{4/3}\right )-4 a^3 b \sqrt [3]{c} d+4 a b^3 c \sqrt [3]{d}+b^4 c^{4/3}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{2/3}}+24 a b^3 d^{2/3} x+3 b^4 d^{2/3} x^2}{6 d^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4/(c + d*x^3),x]

[Out]

(24*a*b^3*d^(2/3)*x + 3*b^4*d^(2/3)*x^2 + (2*Sqrt[3]*(b^4*c^(4/3) + 4*a*b^3*c*d^
(1/3) - 4*a^3*b*c^(1/3)*d - a^4*d^(4/3))*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt
[3]])/c^(2/3) + (2*(b^4*c^(4/3) - 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^(1/3)*d + a^4*d^
(4/3))*Log[c^(1/3) + d^(1/3)*x])/c^(2/3) - ((b^4*c^(4/3) - 4*a*b^3*c*d^(1/3) - 4
*a^3*b*c^(1/3)*d + a^4*d^(4/3))*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/
c^(2/3) + 12*a^2*b^2*d^(2/3)*Log[c + d*x^3])/(6*d^(5/3))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 446, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4/(d*x^3+c),x)

[Out]

1/2*b^4*x^2/d+4*a*b^3*x/d+1/3/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a^4-4/3/d^2/(c/d)^
(2/3)*ln(x+(c/d)^(1/3))*a*b^3*c-1/6/d/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/
3))*a^4+2/3/d^2/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*a*b^3*c+1/3/d/(c/d
)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*a^4-4/3/d^2/(c/d)^(2/3)*
3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*a*b^3*c-4/3/d/(c/d)^(1/3)*ln(x+(
c/d)^(1/3))*a^3*b+1/3/d^2/(c/d)^(1/3)*ln(x+(c/d)^(1/3))*b^4*c+2/3/d/(c/d)^(1/3)*
ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*a^3*b-1/6/d^2/(c/d)^(1/3)*ln(x^2-x*(c/d)^(1/3)
+(c/d)^(2/3))*b^4*c+4/3/d*3^(1/2)/(c/d)^(1/3)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*
x-1))*a^3*b-1/3/d^2*3^(1/2)/(c/d)^(1/3)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*
b^4*c+2*a^2*b^2*ln(d*x^3+c)/d

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(d*x^3 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(d*x^3 + c),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 8.68548, size = 325, normalized size = 1.15 \[ \frac{4 a b^{3} x}{d} + \frac{b^{4} x^{2}}{2 d} + \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{5} - 162 t^{2} a^{2} b^{2} c^{2} d^{4} + t \left (36 a^{7} b c d^{4} + 171 a^{4} b^{4} c^{2} d^{3} + 36 a b^{7} c^{3} d^{2}\right ) - a^{12} d^{4} + 4 a^{9} b^{3} c d^{3} - 6 a^{6} b^{6} c^{2} d^{2} + 4 a^{3} b^{9} c^{3} d - b^{12} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{36 t^{2} a^{3} b c^{2} d^{4} - 9 t^{2} b^{4} c^{3} d^{3} + 3 t a^{8} c d^{4} - 168 t a^{5} b^{3} c^{2} d^{3} + 84 t a^{2} b^{6} c^{3} d^{2} + 26 a^{10} b^{2} c d^{3} + 48 a^{7} b^{5} c^{2} d^{2} - 66 a^{4} b^{8} c^{3} d - 8 a b^{11} c^{4}}{a^{12} d^{4} + 52 a^{9} b^{3} c d^{3} - 52 a^{3} b^{9} c^{3} d - b^{12} c^{4}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4/(d*x**3+c),x)

[Out]

4*a*b**3*x/d + b**4*x**2/(2*d) + RootSum(27*_t**3*c**2*d**5 - 162*_t**2*a**2*b**
2*c**2*d**4 + _t*(36*a**7*b*c*d**4 + 171*a**4*b**4*c**2*d**3 + 36*a*b**7*c**3*d*
*2) - a**12*d**4 + 4*a**9*b**3*c*d**3 - 6*a**6*b**6*c**2*d**2 + 4*a**3*b**9*c**3
*d - b**12*c**4, Lambda(_t, _t*log(x + (36*_t**2*a**3*b*c**2*d**4 - 9*_t**2*b**4
*c**3*d**3 + 3*_t*a**8*c*d**4 - 168*_t*a**5*b**3*c**2*d**3 + 84*_t*a**2*b**6*c**
3*d**2 + 26*a**10*b**2*c*d**3 + 48*a**7*b**5*c**2*d**2 - 66*a**4*b**8*c**3*d - 8
*a*b**11*c**4)/(a**12*d**4 + 52*a**9*b**3*c*d**3 - 52*a**3*b**9*c**3*d - b**12*c
**4))))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215461, size = 427, normalized size = 1.51 \[ \frac{2 \, a^{2} b^{2}{\rm ln}\left ({\left | d x^{3} + c \right |}\right )}{d} + \frac{b^{4} d x^{2} + 8 \, a b^{3} d x}{2 \, d^{2}} - \frac{\sqrt{3}{\left (4 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b^{3} c d - \left (-c d^{2}\right )^{\frac{1}{3}} a^{4} d^{2} - \left (-c d^{2}\right )^{\frac{2}{3}} b^{4} c + 4 \, \left (-c d^{2}\right )^{\frac{2}{3}} a^{3} b d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \, c d^{3}} - \frac{{\left (4 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b^{3} c d - \left (-c d^{2}\right )^{\frac{1}{3}} a^{4} d^{2} + \left (-c d^{2}\right )^{\frac{2}{3}} b^{4} c - 4 \, \left (-c d^{2}\right )^{\frac{2}{3}} a^{3} b d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \, c d^{3}} + \frac{{\left (b^{4} c d^{4} \left (-\frac{c}{d}\right )^{\frac{1}{3}} - 4 \, a^{3} b d^{5} \left (-\frac{c}{d}\right )^{\frac{1}{3}} + 4 \, a b^{3} c d^{4} - a^{4} d^{5}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(d*x^3 + c),x, algorithm="giac")

[Out]

2*a^2*b^2*ln(abs(d*x^3 + c))/d + 1/2*(b^4*d*x^2 + 8*a*b^3*d*x)/d^2 - 1/3*sqrt(3)
*(4*(-c*d^2)^(1/3)*a*b^3*c*d - (-c*d^2)^(1/3)*a^4*d^2 - (-c*d^2)^(2/3)*b^4*c + 4
*(-c*d^2)^(2/3)*a^3*b*d)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(
c*d^3) - 1/6*(4*(-c*d^2)^(1/3)*a*b^3*c*d - (-c*d^2)^(1/3)*a^4*d^2 + (-c*d^2)^(2/
3)*b^4*c - 4*(-c*d^2)^(2/3)*a^3*b*d)*ln(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c*
d^3) + 1/3*(b^4*c*d^4*(-c/d)^(1/3) - 4*a^3*b*d^5*(-c/d)^(1/3) + 4*a*b^3*c*d^4 -
a^4*d^5)*(-c/d)^(1/3)*ln(abs(x - (-c/d)^(1/3)))/(c*d^5)